Pedal Curve

Given a curve $C$, the pedal curve of $C$ with respect to a fixed point $O$ (the Pedal Point) is the locus of the point $P$ of intersection of the Perpendicular from $O$ to a Tangent to $C$. The parametric equations for a curve $(f(t),g(t))$ relative to the Pedal Point $(x_0,y_0)$ are

$$x = {x_0f'^2+f g'^2+(y_0-g)f'g'\over f'^2+g'^2}$$

$$y = {g f'^2+y_0g'^2+(x_0-f)f'g'\over f'^2+g'2^2}.$$

Curve	Pedal Point	Pedal Curve
Astroid	center	Quadrifolium
Cardioid	cusp	Cayley's Sextic
Central Conic	Focus	Circle
Circle	any point	Limaçon
Circle	on Circumference	Cardioid
Circle Involute	center of Circle	Archimedean Spiral
Cissoid of Diocles	Focus	Cardioid
Deltoid	center	Trifolium
Deltoid	cusp	simple Folium
Deltoid	on curve	unsymmetric double folium
Deltoid	Vertex	double folium
Epicycloid	center	Rose
Hypocycloid	center	Rose
Line	any point	point
Logarithmic Spiral	pole	Logarithmic Spiral
Parabola	Focus	Line
Parabola	foot of Directrix	Right Strophoid
Parabola	on Directrix	Strophoid
Parabola	reflection of Focus by Directrix	Maclaurin Trisectrix
Parabola	Vertex	Cissoid of Diocles
Sinusoidal Spiral	pole	Sinusoidal Spiral
Tschirnhausen Cubic	Focus of Pedal	Parabola

See also Negative Pedal Curve

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 46-49 and 204, 1972.

Lee, X. ``Pedal.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Pedal_dir/pedal.html.

Lockwood, E. H. ``Pedal Curves.'' Ch. 18 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 152-155, 1967.

Yates, R. C. ``Pedal Curves.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 160-165, 1952.